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Homomorphisms between mapping class groups

Javier Aramayona and Juan Souto

Geometry & Topology 16 (2012) 2285–2341
Abstract

Suppose that X and Y are surfaces of finite topological type, where X has genus g 6 and Y has genus at most 2g 1; in addition, suppose that Y is not closed if it has genus 2g 1. Our main result asserts that every nontrivial homomorphism Map(X) Map(Y ) is induced by an embedding, ie a combination of forgetting punctures, deleting boundary components and subsurface embeddings. In particular, if X has no boundary then every nontrivial endomorphism Map(X) Map(X) is in fact an isomorphism.

Keywords
mapping class groups, superrigidity
Mathematical Subject Classification 2010
Primary: 20F34
Secondary: 57M07, 20F65
References
Publication
Received: 1 August 2011
Revised: 6 May 2012
Accepted: 27 July 2012
Published: 16 January 2013
Proposed: Cameron Gordon
Seconded: Colin Rourke, Walter Neumann
Authors
Javier Aramayona
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland Galway
University Road
Galway
Ireland
http://www.maths.nuigalway.ie/~javier/
Juan Souto
Mathematics Department
University of British Columbia
1984 Mathematics Road
Vancouver, BC
Canada V6T 1Z2
http://www.math.ubc.ca/~jsouto